Theorem trnfsetN 35442

Description: The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
trnset.a	|- A = ( Atoms ` K )
trnset.s	|- S = ( PSubSp ` K )
trnset.p	|- .+ = ( +P ` K )
trnset.o	|- ._|_ = ( _|_P ` K )
trnset.w	|- W = ( WAtoms ` K )
trnset.m	|- M = ( PAut ` K )
trnset.l	|- L = ( Dil ` K )
trnset.t	|- T = ( Trn ` K )

Assertion

Ref	Expression
trnfsetN	|- ( K e. C -> T = ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) )

Distinct variable groups:   A, d   f, d, q, r, K   f, L   W, q, r

Allowed substitution hints:   A( f, r, q)   C( f, r, q, d)   .+ ( f, r, q, d)   S( f, r, q, d)   T( f, r, q, d)   L( r, q, d)   M( f, r, q, d)   ._|_ ( f, r, q, d)   W( f, d)

Proof of Theorem trnfsetN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. C -> K e. _V )
2		trnset.t	. . 3 |- T = ( Trn ` K )
3		fveq2 6191	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		trnset.a	. . . . . 6 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Atoms ` k ) = A )
6		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Dil ` k ) = ( Dil ` K ) )
7		trnset.l	. . . . . . . 8 |- L = ( Dil ` K )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( Dil ` k ) = L )
9	8	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( Dil ` k ) ` d ) = ( L ` d ) )
10		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( WAtoms ` k ) = ( WAtoms ` K ) )
11		trnset.w	. . . . . . . . 9 |- W = ( WAtoms ` K )
12	10, 11	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( WAtoms ` k ) = W )
13	12	fveq1d 6193	. . . . . . 7 |- ( k = K -> ( ( WAtoms ` k ) ` d ) = ( W ` d ) )
14		fveq2 6191	. . . . . . . . . . . 12 |- ( k = K -> ( +P ` k ) = ( +P ` K ) )
15		trnset.p	. . . . . . . . . . . 12 |- .+ = ( +P ` K )
16	14, 15	syl6eqr 2674	. . . . . . . . . . 11 |- ( k = K -> ( +P ` k ) = .+ )
17	16	oveqd 6667	. . . . . . . . . 10 |- ( k = K -> ( q ( +P ` k ) ( f ` q ) ) = ( q .+ ( f ` q ) ) )
18		fveq2 6191	. . . . . . . . . . . 12 |- ( k = K -> ( _|_P ` k ) = ( _|_P ` K ) )
19		trnset.o	. . . . . . . . . . . 12 |- ._|_ = ( _|_P ` K )
20	18, 19	syl6eqr 2674	. . . . . . . . . . 11 |- ( k = K -> ( _|_P ` k ) = ._|_ )
21	20	fveq1d 6193	. . . . . . . . . 10 |- ( k = K -> ( ( _|_P ` k ) ` { d } ) = ( ._|_ ` { d } ) )
22	17, 21	ineq12d 3815	. . . . . . . . 9 |- ( k = K -> ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) )
23	16	oveqd 6667	. . . . . . . . . 10 |- ( k = K -> ( r ( +P ` k ) ( f ` r ) ) = ( r .+ ( f ` r ) ) )
24	23, 21	ineq12d 3815	. . . . . . . . 9 |- ( k = K -> ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) )
25	22, 24	eqeq12d 2637	. . . . . . . 8 |- ( k = K -> ( ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) <-> ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) ) )
26	13, 25	raleqbidv 3152	. . . . . . 7 |- ( k = K -> ( A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) <-> A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) ) )
27	13, 26	raleqbidv 3152	. . . . . 6 |- ( k = K -> ( A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) <-> A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) ) )
28	9, 27	rabeqbidv 3195	. . . . 5 |- ( k = K -> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } = { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } )
29	5, 28	mpteq12dv 4733	. . . 4 |- ( k = K -> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) = ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) )
30		df-trnN 35393	. . . 4 |- Trn = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) )
31		fvex 6201	. . . . . 6 |- ( Atoms ` K ) e. _V
32	4, 31	eqeltri 2697	. . . . 5 |- A e. _V
33	32	mptex 6486	. . . 4 |- ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) e. _V
34	29, 30, 33	fvmpt 6282	. . 3 |- ( K e. _V -> ( Trn ` K ) = ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) )
35	2, 34	syl5eq 2668	. 2 |- ( K e. _V -> T = ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) )
36	1, 35	syl 17	1 |- ( K e. C -> T = ( d e. A |-> { f e. ( L ` d ) | A. q e. ( W ` d ) A. r e. ( W ` d ) ( ( q .+ ( f ` q ) ) i^i ( ._|_ ` { d } ) ) = ( ( r .+ ( f ` r ) ) i^i ( ._|_ ` { d } ) ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Atomscatm 34550   PSubSpcpsubsp 34782   +Pcpadd 35081   _|_PcpolN 35188   WAtomscwpointsN 35272   PAutcpautN 35273   DilcdilN 35388   TrnctrnN 35389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-trnN 35393

This theorem is referenced by:  trnsetN  35443